## Straight Lines Questions and Answers Part-5

1. A square with each side equal to a lies above the x-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle $\alpha \left(0< \alpha <\pi/4\right)$    with the positive direction of the x-axis. Equation of a diagonal of the square is
a) $y \left(\cos \alpha-\sin \alpha \right)=x\left(\sin\alpha+\cos\alpha\right)$
b) $y \left(\sin\alpha+\cos \alpha \right)+x\left(\cos\alpha-\sin\alpha\right)=a$
c) $x \left(\cos \alpha-\sin \alpha \right)=y\left(\cos\alpha+\sin\alpha\right)$
d) Both a and b

Explanation: Let the side OA make an angle $\alpha$ with the x-axis. Then the coordinates of A are (a cos $\alpha$ , a sin $\alpha$ ). Also, the diagonal OB makes an angle $\alpha$ + $\pi$ /4 with the x-axis, so that its equation is

2. The coordinates of the feet of the perpendiculars from the vertices of a triangle on the opposite sides are (20, 25), (8, 16) and (8, 9). The coordinates of a vertex of the triangle are
a) (5, 10)
b) (50, -5)
c) (15, 30)
d) All of the Above

Explanation: We use the fact that the orthocentre O of the triangle ABC is the incentre of the pedal triangle DEF. Let (h, k) be the coordinates of O.

3.If the area of the triangle formed by the lines y = x, x + y = 2 and the line through P(h, k) and parallel to x-axis is $4h^{2}$, the locus of P can be
a) 2x – y + 1 = 0
b) 2x + y – 1 = 0
c) x – 2y + 1 = 0
d) Both a and b

Explanation: Coordinates of A are (1, 1) which is the point of intersection of the given lines. y = k is the line through P parallel to x-axis which meets the given lines at B and C. So coordinates of B are (k, k) and C are (2 – k, k).

4.Let S be a square with unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c, d denote the lengths of the sides of the quadrilateral, then $\alpha \leq a^{2}+b^{2}+c^{2}+d^{2}\leq\beta$      where
a) $\alpha=1$
b) $\beta=4$
c) $\alpha=2$
d) Both b and c

Explanation: Let the squares of unit area be bounded by the

5. Three lines px + qy + r = 0, qx + ry + p = 0 and rx + py + q = 0 are concurrent if
a) p + q + r = 0
b) $p^{2}+q^{2}+r^{2}=pq+qr+rp$
c) $p^{3}+q^{3}+r^{3}=3pqr$
d) All of the Above

Explanation:

6. An equation of a straight line passing through the point (2, 3) and having an intercept of length 2 units between the straight lines 2x + y = 3, 2x + y = 5 is
a) x – 2 = 0
b) y – 3 = 0
c) 3x + 4y – 18 = 0
d) Both a and c

Explanation: Any line through (2, 3) is

7. A line through the point (a, 0) meets the curve $y^{2}=4ax$     at $P\left(x_{1},y_{1}\right)$   and $Q\left(x_{2},y_{2}\right)$   then
a) $x_{1}x_{2}=a^{2}$
b) $x_{1}x_{2}-y_{1}y_{2}=5a^{2}$
c) $y_{1}y_{2}=-4a^{2}$
d) All of the Above

Explanation: Let the equation of the line through (a, 0) be y = m(x – a), which meets the curve y2= 4ax at points for which m2(x – a)2 = 4ax

8. If the two lines represented by $x^{2}\left(\tan^{2} \theta+\cos^{2}\theta\right)-2xy \tan\theta+y^{2}\sin^{2}\theta=0$
make angles $\alpha,\beta$   with the x-axis, then
a) $\tan\alpha+\tan\beta=4cosec2\theta$
b) $\frac{\tan\alpha}{\tan\beta}=\frac{2+\sin2\theta}{2-\sin2\theta}$
c) $\tan\alpha-\tan\beta=2$
d) All of the Above

Explanation: Let the lines represented by the given equation be

9. If two of the lines given by $3x^3+3x^2y-3xy^2+dy^3=0$       are at right angles then the slope of one of them is
a) -1
b) 1
c) 3
d) Both a and b

10. $9x^2+2hxy+4y^2+6x+2fy-3=0$        represent two parallel lines if